# Optimal a priori error estimates of parabolic optimal control problems with a moving point control

## TL;DR

This paper establishes optimal a priori error estimates for a parabolic optimal control problem involving a moving point source, correcting previous flawed analysis and providing new error bounds with logarithmic factors.

## Contribution

It offers the first correct proof of optimal error estimates for this problem, including global and local error analysis on a curve, improving upon prior flawed results.

## Key findings

- Optimal convergence rates achieved in discretization
- Error estimates include logarithmic factors
- Corrected proof addresses previous flaws in analysis

## Abstract

In this paper we consider a parabolic optimal control problem with a Dirac type control with moving point source in two space dimensions. We discretize the problem with piecewise constant functions in time and continuous piecewise linear finite elements in space. For this discretization we show optimal order of convergence with respect to the time and the space discretization parameters modulo some logarithmic terms. Error analysis for the same problem was carried out in the recent paper [17], however, the analysis there contains a serious flaw. One of the main goals of this paper is to provide the correct proof. The main ingredients of our analysis are the global and local error estimates on a curve, that have an independent interest.